Building on the concept of FIQCP, the quantum criticality in the presence of external fluctuations provided by the Dirac fermions
has sparked tremendous interest[1]}, [2]}, [3]}, [4]}.
A representative example is the semimetal-insulator transition of interacting electrons on the honeycomb lattice[2]}, [4]}.
When the interaction is sufficiently strong, the electrons may undergo continuous transition from semimetallic phase to a symmetry broken insulating phase characterized by an ordered ground state[7]}. A large on-site repulsive interaction, for example, gives rise to a spin-density-wave state[8]}, [9]},
whereas a nearest-neighbor interaction would induce a charge-density-wave state[10]}.
A large number of interacting Dirac fermion model exhibit the continuous phase transition,
e.g., spin liquid[11]}, superconducting and XY phases [12]}, [13]}, [14]}, [15]}, [16]},
and exotic topological phases[17]}, [18]}.
Moreover, the recent theoretical studies suggest the multicritical point may be achieved in three ways:
(i) condensation of topological defects[19]},
(ii) anticommuting mass terms[20]},
(iii) interaction instabilities[21]}, [22]}, [12]}.
It's argued that the quantum criticality occurs in the vicinity of these critical points and the universality class are defined through different microscopic models. Since the presence of external fluctuations provided by gapless Dirac fermions, these critical points are not captured by the conventional \(O(N)\) 
universality classes. Instead, they are described by the Gross-Neveu-Yukawa theory, defining qualitatively different chiral Gross-Neveu-Yukawa (GNY) universality classes[24]}.
Indeed, the critical behavior of a large number of transitions in condensed matter systems are captured by the GNY model.
To describe the critical behavior of GNY (or the gauged QED-GNY) theory, various theoretical methods and numerical methods have been applied, i.e., perturbative RG[25]}, [10]}, [27]}, [28]}, [29]},
nonperturbative functional RG[30]}, [8]}, large-N method[32]}, [9]} and quantum Monte Carlo simulations[3]}, [35]}, [36]}, [37]}.
In view of the symmetry of the broken phase, the GNY universality class comprises
chiral Ising (\(\mathbb {Z}_2\) ) class [24]}, [39]}, [25]}, [28]}, [10]},
chiral XY [O(2)] class[43]}, [44]}, [45]}, [3]}, [30]}, [48]}
and chiral Heisenberg[SU(2)] universality class[8]}, [32]}, [25]}, [9]}, [16]}.
